Shock and wear degradating systems under three types of repair

We study a general shock and wear model under different types of repair applying matrix-analytic methods. The shocks arrive following a Markovian arrival process. The lifetime of the system follows a phase-type distribution, in which good and bad states are well-differenced. The system can fail from these two sets of states. Three repairs are considered, depending on the set where the system returns after repair: same as good/same as old, bad as old, and improving repair. A general system governed by a multidimensional Markov process is constructed, the generator calculated, and the stationary distribution, the availability, and the rate of occurrence of failures are expressed in algorithmic form. The three classes of repairs are explicitly deduced from a general system. Moreover, a system similar to the general one with a limited number N of repairs is studied, calculating the same performance measures as in the previous system and the renewal process associated to the replacements after the arrival of the (N + 1)th failure is determined. A numerical application is performed for illustrating the calculations.